Submitted to the Annals of Applied Statistics

Submitted to the Annals of Applied Statistics

HOW OFTEN DOES THE BEST TEAM WIN?A UNIFIED APPROACH TO UNDERSTANDING RANDOMNESS

IN NORTH AMERICAN SPORT

By Michael J. Lopez

Skidmore Collegeand

By Gregory J. MatthewsLoyola University Chicago

and

By Benjamin S. BaumerSmith College

Statistical applications in sports have long centered on how tobest separate signal (e.g. team talent) from random noise. However,most of this work has concentrated on a single sport, and the devel-opment of meaningful cross-sport comparisons has been impeded bythe difficulty of translating luck from one sport to another. In thismanuscript, we develop Bayesian state-space models using bettingmarket data that can be uniformly applied across sporting organiza-tions to better understand the role of randomness in game outcomes.These models can be used to extract estimates of team strength,the between-season, within-season, and game-to-game variability ofteam strengths, as well each team’s home advantage. We implementour approach across a decade of play in each of the National FootballLeague (NFL), National Hockey League (NHL), National BasketballAssociation (NBA), and Major League Baseball (MLB), finding thatthe NBA demonstrates both the largest dispersion in talent and thelargest home advantage, while the NHL and MLB stand out for theirrelative randomness in game outcomes. We conclude by proposingnew metrics for judging competitiveness across sports leagues, bothwithin the regular season and using traditional postseason tourna-ment formats. Although we focus on sports, we discuss a number ofother situations in which our generalizable models might be usefullyapplied.

1. Introduction. Most observers of sport can agree that game outcomes are tosome extent subject to chance. The line drive that miraculously finds the fielder’sglove, the fumble that bounces harmlessly out-of-bounds, the puck that ricochetsinto the net off of an opponent’s skate, or the referee’s whistle on a clean blockcan all mean the difference between winning and losing. Yet game outcomes are notcompletely random—there are teams that consistently play better or worse than the

Keywords and phrases: sports analytics, Bayesian modeling, competitive balance, MCMC

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average team. To what extent does luck influence our perceptions of team strengthover time?

One way in which statistics can lead this discussion lies in the untangling of signaland noise when comparing the caliber of each league’s teams. For example, is team ibetter than team j? And if so, how confident are we in making this claim? Central tosuch an understanding of sporting outcomes is that if we know each team’s relativestrength, then, a priori, game outcomes—including wins and losses—can be viewed asunobserved realizations of random variables. As a simple example, if the probabilitythat team i beats team j at time k is 0.75, this implies that in a hypothetical infinitenumber of games between the two teams at time k, i wins three times as often asj. Unfortunately, in practice, team i will typically only play team j once at time k.Thus, game outcomes alone are unlikely to provide enough information to preciselyestimate true probabilities, and, in turn, team strengths.

Given both national public interest and an academic curiosity that has extendedacross disciplines, many innovative techniques have been developed to estimate teamstrength. These approaches typically blend past game scores with game, team, andplayer characteristics in a statistical model. Corresponding estimates of talent areoften checked or calibrated by comparing out-of-sample estimated probabilities ofwins and losses to observed outcomes. Such exercises do more than drive water-coolerconversation as to which team may be better. Indeed, estimating team rankings hasdriven the development of advanced statistical models (Bradley and Terry, 1952;Glickman and Stern, 1998) and occasionally played a role in the decision of whichteams are eligible for continued postseason play (CFP, 2014).

However, because randomness manifests differently in different sports, a limitationof sport-specific models is that inferences cannot generally be applied to other com-petitions. As a result, researchers who hope to contrast one league to another oftenfocus on the one outcome common to all sports: won-loss ratio. Among other flaws,measuring team strength using wins and losses performs poorly in a small samplesize, ignores the game’s final score (which is known to be more predictive of futureperformance than won-loss ratio (Boulier and Stekler, 2003)), and is unduly impactedby, among other sources, fluctuations in league scheduling, injury to key players, andthe general advantage of playing at home. In particular, variations in season lengthbetween sports—NFL teams play 16 regular season games each year, NHL and NBAteams play 82, while MLB teams play 162—could invalidate direct comparisons ofwin percentages alone. As an example, the highest annual team winning percentageis roughly 87% in the NFL but only 61% in MLB, and part (but not all) of thatdifference is undoubtedly tied to the shorter NFL regular season. As a result, untilnow, analysts and fans have never quite been able to quantify inherent differencesbetween sports or sports leagues with respect to randomness and the dispersion andevolution of team strength. We aim to fill this void.

In the sections that follow, we present a unified and novel framework for the si-multaneous comparison of sporting leagues, which we implement to discover inherentdifferences in North American sport. First, we validate an assumption that game-

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RANDOMNESS IN SPORT 3

level probabilities provided by betting markets provide unbiased and low-varianceestimates of the true probabilities of wins and losses in each professional contest. Sec-ond, we extend Bayesian state-space models for paired comparisons (Glickman andStern, 1998) to multiple domains. These models use the game-level betting marketprobabilities to capture implied team strength and variability. Finally, we presentunique league-level properties that to this point have been difficult to capture, andwe use the estimated posterior distributions of team strengths to propose novel met-rics for assessing league parity, both for the regular season and postseason. We findthat, on account of both narrower distributions of team strengths and smaller homeadvantages, a typical contest in the NHL or MLB is much closer to a coin-flip thanone in the NBA or NFL.

1.1. Literature review. The importance of quantifying team strength in competi-tion extends across disciplines. This includes contrasting league-level characteristicsin economics (Leeds and Von Allmen, 2004), estimating game-level probabilities instatistics (Glickman and Stern, 1998), and classifying future game winners in fore-casting (Boulier and Stekler, 2003). We discuss and synthesize these ideas below.

1.1.1. Competitive balance. Assessing the competitive balance of sports leagues isparticularly important in economics and management (Leeds and Von Allmen, 2004).While competitive balance can purportedly measure several different quantities, ingeneral it refers to levels of equivalence between teams. This could be equivalencewithin one time frame (e.g. how similar was the distribution of talent within a sea-son?), between time frames (e.g. year-to-year variations in talent), or from the be-ginning of a time frame until the end (e.g. the likelihood of each team winning achampionship at the start of a season).

The most widely accepted within-season competitive balance measure is Noll-Scully(Noll, 1991; Scully, 1989). It is computed as the ratio of the observed standard de-viation in team win totals to the idealized standard deviation, which is defined asthat which would have been observed due to chance alone if each team were equalin talent. Larger Noll-Scully values are believed to reflect greater imbalance in teamstrengths.

While Noll-Scully has the positive quality of allowing for interpretable cross-sportcomparisons, a reliance on won-loss outcomes entails undesireable properties as well(Owen, 2010; Owen and King, 2015). For example, Noll-Scully increases, on average,with the number of games played (Owen and King, 2015), hindering any compar-isons of the NFL (16 games) to MLB (162), for example. Additionally, each of theleagues employ some form of an unbalanced schedule. Teams in each of MLB, theNBA, NFL, and NHL play intradivisional opponents more often than interdivisionalones, and intraconference opponents more often than interconference ones, meaningthat one team’s won-loss record may not be comparable to another team’s due todifferences in the respective strengths of their opponents (Lenten, 2015). Moreover,the NFL structures each season’s schedule so that teams play interdivisional games



against opponents that finished with the same division rank in the standings in theprior year. In expectation, this punishes teams that finish atop standings with toughergames, potentially driving winning percentages toward 0.500. Unsurprisingly, unbal-anced scheduling and interconference play can lead to imprecise competitive balancemetrics derived from winning percentages (Utt and Fort, 2002). As one final weak-ness, varying home advantages between sports leagues, as shown in Moskowitz andWertheim (2011), could also impact comparisons of relative team quality that arepredicated on wins and losses.

Although metrics for league-level comparisons have been frequently debated, theimportance of competitive balance in sports is more uniformly accepted, in large partdue to the uncertainty of outcome hypothesis (Rottenberg, 1956; Knowles, Sheronyand Haupert, 1992; Lee and Fort, 2008). Under this hypothesis, league success—asjudged by attendance, engagement, and television revenue—correlates positively withteams having equal chances of winning. Outcome uncertainty is generally consideredon a game-level basis, but can also extend to season-level success (i.e, teams havingequivalent chances at making the postseason). As a result, it is in each league’s bestinterest to promote some level of parity—in short, a narrower distribution of teamquality—to maximize revenue (Crooker and Fenn, 2007). Related, the Hirfindahl-Hirschman Index (Owen, Ryan and Weatherston, 2007) and Competitive BalanceRatio (Humphreys, 2002) are two metrics attempting to quantify the relative chancesof success that teams have within or between certain time frames.

1.1.2. Approaches to estimating team strength. Competitive balance and outcomeuncertainty are rough proxies for understanding the distribution of talent amongteams. For example, when two teams of equal talent play a game without a homeadvantage, outcome uncertainty is maximized; e.g., the outcome of the game is equiv-alent to a coin flip. These relative comparisons of team strength began in statisticswith paired comparison models, which are generally defined as those designed to cal-ibrate the equivalence of two entities. In the case of sports, the entities are teams orindividual athletes.

The Bradley-Terry model (BTM, Bradley and Terry (1952)) is considered to be thefirst detailed paired comparison model, and the rough equivalent of the soon thereafterdeveloped Elo rankings (Elo, 1978; Glickman, 1995). Consider an experiment with ttreatment levels, compared in pairs. BTM assumes that there is some true orderingof the probabilities of efficacy, π1, . . . , πt, with the constraints that

∑ti=1 πi = 1 and

πi ≥ 0 for i = 1, . . . , t. When comparing treatment i to treatment j, the probabilitythat treatment i is preferable to j (i.e. a win in a sports setting) is computed as πi

πi+πj.

Glickman and Stern (1998) and Glickman and Stern (2016) build on the BTM byallowing team-strength estimates to vary over time through the modeling of pointdifferential in the NFL, which is assumed to follow an approximately normal distribu-tion. Let y(s,k)ij be the point differential of a game during week k of season s betweenteams i and j. In this specification, i and j take on values between 1 and t, where tis the number of teams in the league. Let θ(s,k)i and θ(s,k)j be the strengths of teams



i and j, respectively, in season s during week k, and let αi be the home advantageparameter for team i for i = 1, . . . , t. Glickman and Stern (1998) assume that for agame played at the home of team i during week k in season s,

E[y(s,k)ij |θ(s,k)i, θ(s,k)j , αi] = θ(s,k)i − θ(s,k)j + αi,

where E[y(s,k)ij |θ(s,k)i, θ(s,k)j , αi] is the expected point differential given i and j’s teamstrengths and the home advantage of team i.

The model of Glickman and Stern (1998) allows for team strength parameters tovary stochastically in two distinct ways: from the last week of season s to the firstweek of season s+ 1, and from week k of season s to week k+ 1 of season s. As such,it is termed a ‘state-space’ model, whereby the data is a function of an underlyingtime-varying process plus additional noise.

Glickman and Stern (1998) propose an autoregressive process to model team strengths,whereby over time, these parameters are pulled toward the league average. Underthis specification, past and future season performances are incorporated into season-specific estimates of team quality. Perhaps as a result, Koopmeiners (2012) identifiesbetter fits when comparing state-space models to BTM’s fit separately within eachseason. Additionally, unlike BTM’s, state-space models would not typically suffer fromidentifiability problems were a team to win or lose all of its games in a single season(a rare, but extant possibility in the NFL).1 For additional and related state-spaceresources, see Fahrmeir and Tutz (1994), Knorr-Held (2000), Cattelan, Varin andFirth (2013), Baker and McHale (2015), and Manner (2015). Additionally, Matthews(2005), Owen (2011), Koopmeiners (2012), Tutz and Schauberger (2015), and Wolfsonand Koopmeiners (2015) implement related versions of the original BTM.

Although the state-space model summarized above appears to work well in theNFL, a few issues arise when extending it to other leagues. First, with point differen-tial as a game-level outcome, parameter estimates would be sensitive to the relativeamount of scoring in each sport. Thus, comparisons of the NHL and MLB (wheregames, on average, are decided by a few goals or runs) to the NBA and NFL (wheregames, on average, are decided by about 10 points) would require further scaling.Second, a normal model of goal or run differential would be inappropriate in low scor-ing sports like hockey or baseball, where scoring outcomes follow a Poisson process(Mullet, 1977; Thomas et al., 2007). Finally, NHL game outcomes would entail anextra complication, as roughly 25% of regular season games are decided in overtimeor a shootout.

In place of paired comparison models, alternative measures for estimating teamstrength have also been developed. Massey (1997) used maximum likelihood estima-tion and American football outcomes to develop an eponymous rating system. A moregeneral summary of other rating systems for forecasting use is explored by Boulierand Stekler (2003). In addition, support vector machines and simulation models have

1In the NFL, the 2007 New England Patriots won all of their regular season games, while the 2008Detroit Lions lost all of their regular season games.



been proposed in hockey (Demers, 2015; Buttrey, 2016), neural networks and naıveBayes implemented in basketball (Loeffelholz et al., 2009; Miljkovic et al., 2010), lin-ear models and probit regressions in football (Harville, 1980; Boulier and Stekler,2003), and two stage Bayesian models in baseball (Yang and Swartz, 2004). Whilethis is a non-exhaustive list, it speaks to the depth and variety of coverage that sportsprediction models have generated.

1.2. Betting market probabilities. In many instances, researchers derive estimatesof team strength in order to predict game-level probabilities. Betting market informa-tion has long been recommended to judge the accuracy of these probabilities (Harville,1980; Stern, 1991). Before each contest, sports books—including those in Las Vegasand in overseas markets—provide a price for each team, more commonly known asthe money line.

Mathematically, if team i’s money line is ì against team j (with correspondingmoney line `j), where |ì| ≥ 100, then the boundary win probability for that team,pi(ì), is given by:

pi(ì) =

{100

100+ìif ì ≥ 100

|ì|100+|ì| if ì ≤ −100

.

The boundary win probability represents the threshold at which point betting onteam i would be profitable in the long run.

As an example, suppose the Chicago Cubs were favored (ì = −127 on the moneyline) to beat the Arizona Diamondbacks (`j = 117). The boundary win probabilityfor the Cubs would be pi(−127) = 0.559; for the Diamondbacks, pj(117) = 0.461.Boundary win probabilities sum to greater than one by an amount collected by thesportsbook as profit (known colloquially as the “vig” or “vigorish”). However, it isstraightforward to normalize boundary probabilities to sum to unity to estimate pij ,the implied probability of i defeating j:

pij =pi(ì)

pi(ì) + pj(`j).(1)

In our example, dividing each boundary probability by 1.02 = (0.559 + 0.461) implieswin probabilities of 54.8% for the Cubs and 45.2% for the Diamondbacks.

In principle, money line prices account for all determinants of game outcomes knownto the public prior to the game, including team strength, location, and injuries. Acrosstime and sporting leagues, researchers have identified that it is difficult to estimatewin probabilities that are more accurate than the market; i.e, the betting marketsare efficient. As an incomplete list, see Harville (1980); Gandar et al. (1988); Lacey(1990); Stern (1991); Carlin (1996); Colquitt, Godwin and Caudill (2001); Spannand Skiera (2009); Nichols (2012); Paul and Weinbach (2014); Lopez and Matthews(2015). Interestingly, Colquitt, Godwin and Caudill (2001) suggested that the effi-ciency of college basketball markets was proportional to the amount of pre-gameinformation available—with the amount known about professional sports teams, this



would suggest that markets in the NFL, NBA, NHL and MLB are as efficient as theycome. Manner (2015) merged predictions from a state-space model with those frombetting markets, finding that the combination of both predictions only occasionallyoutperformed betting markets alone.

We are not aware of any published findings that have compared leagues usingmarket probabilities. Given the varying within-sport metrics of judging team qualityand the limited between-sport approaches that rely on wins and losses alone, we aimto extend paired comparison models using money line information to better capturerelative team equivalence in a method that can be applied generally.

2. Validation of betting market data. We begin by confirming the accuracyof betting market data with respect to game outcomes. Regular season game resultand betting line data in the four major North American professional sports leagues(MLB, NBA, NFL, and NHL) were obtained for a nominal fee from Sports Insights(https://www.sportsinsights.com). Although these game results are not official,they are accurate and widely-used. Our models were fit to data from the 2006–2016seasons, except for the NFL, in which the 2016 season was not yet completed.

These data were more than 99.3% complete in each league, in the sense that thereexisted a valid betting line for nearly all games in these four sports across this timeperiod. Betting lines provided by Sports Insights are expressed as payouts, whichwe subsequently convert into implied probabilities. The average vig in our data setis 1.93%, but is always positive, resulting in revenue for the sportsbook over a longrun of games. In circumstances where more than one betting line was available for aparticular game, we included only the line closest to the start time of the game. Asummary of our data is shown in Table 1.

Sport (q) tq ngames pgames nbets pbets Coverage

MLB 30 26728 0.541 26710 0.548 0.999NBA 30 13290 0.595 13245 0.615 0.997NFL 32 2560 0.563 2542 0.589 0.993NHL 30 13020 0.548 12990 0.565 0.998

Table 1Summary of cross-sport data. tq is the number of unique teams in each sport q. ngames records thenumber of actual games played, while nbets records the number of those games for which we have abetting line. pgames is the mean observed probability of a win for the home team, while pbets is the

mean implied probability of a home win based on the betting line. Note that we have near totalcoverage (betting odds for almost every game) across all four major sports.

We also compared the observed probabilities of a home win to the correspondingprobabilities implied by our betting market data (Figure 1). In each of the four sports,Hosmer-Lemeshow tests of an efficient market hypothesis using 10 equal-sized bins ofgames did not show evidence of a lack of fit when comparing the number of observedand expected wins in each bin. Thus, we find no evidence to suggest that the prob-abilities implied by our betting market data are biased or inaccurate—a conclusionthat is supported by the body of academic literature referenced above. Accordingly,


https://www.sportsinsights.com


we interpret these probabilities as “true.”

3. Bayesian state-space model. Our model below expands the state-spacespecification provided by Glickman and Stern (1998) to provide a unified frameworkfor contrasting the four major North American sports leagues.

Let p(q,s,k)ij be the probability that team i will beat team j in season s during weekk of sports league q, for q ∈ {MLB,NBA,NFL,NHL}. The p(q,s,k)ij ’s are assumed tobe known, calculated using sportsbook odds via Equation (1). In using game prob-abilities, we have a cross-sport outcome that provides more information than onlyknowing which team won the game or what the score was.

In our notation, i, j = 1, . . . , tq, where tq is the number of teams in sport q suchthat tMLB = tNBA = tNHL = 30 and tNFL = 32. Additionally, s = 1, . . . , Sq andk = 1, . . . ,Kq, where Sq and Kq are the number of seasons and weeks, respectively inleague q. In our data, KNFL = 17, KNBA = 25, KMLB = KNHL = 28, with SNFL = 10and SMLB = SNBA = SNHL = 11.

Our next step in building a model specifies the home advantage, and one immediatehurdle is that in addition to having different numbers of teams in each league, certainfranchises may relocate from one city to another over time. In our data set, there weretwo relocations, Seattle to Oklahoma City (NBA, 2008) and Atlanta to Winnipeg(NHL, 2011). Let αq0 be the league-wide home advantage (HA) in league q, and letα(q)i? be the team specific effect (positive or negative) for team i among games playedin city i?, for i? = 1, . . . , t?q . Here, t?q is the total number of home cities; in our data,t?MLB = 30, t?NBA = t?NHL = 31, and t?NFL = 32.

Letting θ(q,s,k)i and θ(q,s,k)j be season-week team strength parameters for teams iand j, respectively, we assume that

E[logit(p(q,s,k)ij)|θ(q,s,k)i, θ(q,s,k)j , αq0 , α(q)i? ] = θ(q,s,k)i − θ(q,s,k)j + αq0 + α(q)i? ,

where logit(.) is the log-odds transform. Note that θ(q,s,k)i and θ(q,s,k)j reflect absolutemeasures of team strength, and translate into each team’s probability of beating aleague average team. We center team strength and individual home advantage esti-mates about 0 to ensure that our model is identifiable (e.g.,

∑tqi=1 θ(q,s,k)i = 0 for all

q, s, k and∑t?q

i?=1 α(q)i? = 0 )Let p(q,s,k) represent the vector of length g(q,s,k), the number of games in league

q during week k of season s, containing all of league q’s probabilities in week k ofseason s. Our first model of game outcomes, henceforth referred to as the individualhome advantage model (Model IHA), assumes that

logit(p(q,s,k)) ∼ N(θ(q,s,k)X(q,s,k) + αq0Jg(q,s,k) +αααqZ(q,s,k), σ2q,gameIg(q,s,k)),

where θ(q,s,k) is a vector of length tq containing the team strength parameters in season

s during week k and αααq ={α(q)1, . . . , α(q)t?q

}. Note that αααq does not vary over time



NFL NHL

MLB NBA

0% 25% 50% 75% 100% 0% 25% 50% 75% 100%

0%

25%

50%

75%

100%

0%

25%

50%

75%

100%

Betting Market Estimated Probability of Home Win

Obs

erve

d P

roba

bilit

y of

Hom

e W

in

N5001000

Fig 1. Accuracy of probabilities implied by betting markets. Each dot represents a bin of impliedprobabilities rounded to the nearest hundredth. The size of each dot (N) is proportional to the number ofgames that lie in that bin. We note that across all four major sports, the observed winning percentagesaccord with those implied by the betting markets. The dotted diagonal line indicates a completely fairmarket where probabilities from the betting markets correspond exactly to observed outcomes. In eachsport, Hosmer-Lemeshow tests suggest that an efficient market hypothesis cannot be rejected.



(i.e. HA is assumed to be constant for a team over weeks and seasons). X(q,s,k) andZ(q,s,k) contain g(q,s,k) rows and tq and t?q columns, respectively. The matrix X(q,s,k)

contains the values {1, 0,−1} where for a given row (i.e. one game) the value of ith

column in that row is a 1/-1 if the ith team played at home/away in the given gameand 0 otherwise. Z(q,s,k) is a matrix containing a 1 in column i? if the correspondinggame was played in city i?, and 0 otherwise. Finally, σ2q,game is the game-level vari-ance, Jg(q,s,k) is a column vector of length g(q,s,k) containing all 1’s, and Ig(q,s,k) is anidentity matrix with dimension g(q,s,k) × g(q,s,k).

In addition, we propose a simplified version of Model IHA, labelled as Model CHA(constant home advantage), which assumes that the HA within each sport is identicalfor each franchise, such that

logit(p(q,s,k)) ∼ N(θ(q,s,k)X(q,s,k) + αq0Jg(q,s,k) , σ2q,gameIg(q,s,k)).

In Model CHA, matrices p(q,s,k), X(q,s,k), Jg(q,s,k) , and Ig(q,s,k) are specified identicallyto Model IHA. As a result, for a game between home team i and away team j duringweek k of season s, E[logit(p(q,s,k)ij)] = θ(q,s,k)i − θ(q,s,k)j + αq0 under Model CHA.

Similar to Glickman and Stern (1998), we allow the strength parameters of theteams to vary auto-regressively from season-to-season and from week-to-week. In gen-eral, this entails that team strength parameters are shrunk towards the league averageover time in expectation. Formally,

θ(q,s+1,1)|θq,s,Kq , γq,season, σ2q,season ∼ N(γq,seasonθ(q,s,Kq), σ

2q,seasonItq)

for all s ∈ 1, . . . , Sq − 1, and

θ(q,s,k+1)|θ(q,s,k), γq,week, σ2q,week ∼ N(γq,weekθ(q,s,k), σ2q,weekItq)

for all s ∈ 1, . . . , Sq, k ∈ 1, . . . ,Kq − 1.In this specification, γq,week is the autoregressive parameter from week-to-week,

γq,season is the autoregressive parameter from season-to-season, and Itq is the identitymatrix of dimension tq × tq.

Given the time-varying nature of our specification, all specifications use a Bayesianapproach to obtain model estimates. For sport q, the team strength parameters forweek k = 1 and season s = 1 have a prior distribution of

θ(q,1,1)i ∼ N(0, σ2q,season) , for all i ∈ 1, . . . , tq.

Team specific home advantage parameters have a similar prior, namely,

α(q)i? ∼ N(0, σ2q,α) , for i ∈ 1, . . . , t?q .



Finally, letting τ2q,game = 1/σ2q,game, τ2q,season = 1/σ2q,season, τ2q,week = 1/σ2q,week, and

τ2q,α = 1/σ2q,α, we assume the following prior distributions (Gelman et al., 2006):

τ2q,game ∼ Uniform(0, 1000) αq0 ∼ N(0, 10000)

τ2q,season ∼ Uniform(0, 1000) γq,season ∼ Uniform(0, 1)

τ2q,week ∼ Uniform(0, 1000) γq,week ∼ Uniform(0, 1.5)

τ2q,α ∼ Uniform(0, 1000)

Note that we cap γq,week and γq,season at 1.5 and 1.0, respectively, corresponding toprior beliefs in whether or not team strengths could explode within (unlikely, butfeasible) or between (highly unlikely) seasons.

One of our main interests lies in gauging the game-level equivalence of each league’steams; i.e., how likely was it or will it be for each team to beat other teams? In thisrespect, we are interested in both looking backwards across time (descriptive) as wellas looking forwards (predictive). However, Models IHA and CHA each blend outcomesfrom weeks prior to, during, and after week k to estimate team strength. While thisis ideal for measuring league parity looking backwards, it is less appropriate to makefuture game predictions. As such, in each q for season Sq (the last season of our data),we fit a series of state-space models using Model IHA, done on a weekly basis (theseare termed sequential fits, as opposed to cumulative). Formally, for k = 2, . . . ,Kq inseason Sq, we fit Model IHA only on games during k or prior. Sequential fits can beused to provide a sense of the predictive capability of our model.

Posterior distributions of each parameter are estimated using Markov Chain MonteCarlo (MCMC) methods. We use Gibbs sampling via the rjags package (Plummer,2016) in the R (R Core Team, 2016) statistical computing environment to obtainposterior distributions, separately for each q.2 Three chains—using 40,000 iterationsafter a burn-in of 4,000 draws, fit with a thin of 5 —yield 8,000 posterior samplesin each q.3 Visual inspection of trace plots with parallel chains are used to confirmconvergence. To assess the underlying assumptions of Models IHA and CHA, includingour use of the logit transform on our probability outcomes, we use posterior predictivedistribution checks, as in Gelman et al. (2014). Comparisons of Models IHA and CHAare made using the Deviance Information Criterion (DIC, Spiegelhalter et al. (2002))and by examining each model’s posterior predictive distribution.

While we are unable to share the exact betting market data due to licensing re-strictions, a simplified version of our game-level data, the data wrangling code, Gibbssampling code, posterior draws, and the code used to obtain posterior estimates andfigures are all posted to a GitHub repository, available at https://github.com/

bigfour/competitiveness.

2Alternatively, we could have fit one model and pooled information across sports. Given the largebetween-league differences in structure, we opt against this approach.

32000 iterations were used for sequential fits with a burn-in of 1000.


https://github.com/bigfour/competitiveness

https://github.com/bigfour/competitiveness


4. Model Assessment. We begin by validating and comparing the fits of ModelsIHA and CHA.

4.1. Model fit. Trace plots of αq0 , γq,season, γq,week, σq,game, σq,season, and σq,weekare shown for each q in Figures 9–12 in the Appendix. Visual inspection of theseplots does not provide evidence of a lack of convergence or of autocorrelation betweendraws. These trace plots stem from Model IHA; conclusions are similar when plottingdraws from Model CHA.

Table 2 shows the deviance information criterion (DIC) for each fit in each league,along with the difference in DIC values and the associated standard error (SE). Ineach of the NHL, NBA, and NFL, fits with a team-specific HA (Model IHA) yieldedlower DIC’s (lower is better) by a statistically meaningful margin, with the mostnoticeable difference in fit improvement in the NBA. DIC’s were also lower in MLBusing Model IHA, although differences were not significant.

Model IHA Model CHA Difference (SE)

MLB -8538 -8481 -56.8 (37.9)NBA 6864 7188 -323.9 (40.5)NFL 1135 1288 -153.2 (24.3)NHL -18294 -18128 -165.8 (37.7)

Table 2Deviance information criterion (DIC) by sport and model, along with the difference in DIC and theassociated standard errors (SE, in parentheses). IHA: individual home advantage, CHA: constant

home advantage

These results suggest that chance alone likely does not account for observed differ-ences in the home advantage among teams in the NBA, NHL, and NFL. For the NFL,this implication matches the findings of Glickman and Stern (1998), who identifiedmeaningful between-franchise differences in terms of playing at home. For consistency,results that follow use model estimates from Model IHA.

4.2. Posterior predictive checks. We next address the fit of Models IHA and CHAby looking at the posterior predictive distribution of each. Formally, we assess whetherModels IHA and CHA can use draws from their respective posterior distributions togenerate game-level data that roughly matches the observed data.

Our specific interest lies in the posterior predictive distribution of the logit of im-plied probabilities, p(logit(p(q,s,k))|logit(p(q,s,k))). To draw values, we randomly sam-ple from the joint posterior distribution of the parameters (i.e. team strength, homefield advantage, and variance parameters). Then, conditional on the drawn parame-ters, we randomly draw from the distribution of logit(p(q,s,k)). Recall that in the IHAmodel, this distribution is assumed to be normal with the following form:

logit(p(q,s,k)) ∼ N(θ(q,s,k)X(q,s,k) + αq0Jg(q,s,k) +αααqZ(q,s,k), σ2q,gameIg(q,s,k)).

We used 20 simulated sets of logit probabilities from this posterior distribution, aswell as 20 more from the posterior distribution of Model CHA.



Figure 2 overlays each of Model IHA’s 20 posterior predictive distributions of logitprobabilities (shown in gray density curves) along with the observed distribution oflogit probabilities (shown in red). By and large, the observed distributions of logitprobabilities are similar to the simulated distributions in each sport. In particular, thedensity in the tails of the posterior predictive distributions (reflecting probabilitiesnear 0 or 1) does not show any meaningful departure from the observed distributions.

We purposefully use a lower bandwith for the density curves in Figure 2 to highlightinteresting discrepancies between the observed and predictive distributions. In theNBA and NFL, for example, the observed distribution is slightly lower than thesimulated distributions with logit probabilities near 0 (i.e., both teams have a winprobability of 0.5). This is likely occurring due to preference of sportsbooks to setprices that are rounded to the nearest 5 (e.g. -105, -110, -155, etc.). As an example,there are 33 NFL games where the home team’s boundary price is -185 (1.3% ofgames), and there are 22 other prices that are observed for the home team in 15 ormore unique games. Given that Models CHA and IHA do not extract back to roundedprices for each team, it is not surprising that our posterior predictive distributionsare smoother than the observed data. Similarly, Glickman and Stern (1998) founddiscrepancies between the observed distribution of point differential in the NFL andthe posterior predictive distributions of point differential, on account of the increasedlikelihood of games ending with margins of victory of 3 or 7 in the NFL. We believethat we are observing a similar phenomenon, but based on the increased likelihoodof a sportsbook to assign rounded odds.

Next, we use posterior predictive distributions to compare the appropriateness ofModels IHA and CHA for each team, as well as to contrast each of the two modelsto one another. To do this, we calculate the average discrepancy between the meanposterior predictive distribution of each game and the observed game probability,averaged over home team for each model. These team level results are shown inFigure 3. Discrepencies from Model CHA are shown in via circles, with arrows pointingtowards the average discrepency for Model IHA. The color of the arrow (blue for yes,red for no) identifies whether, on average, Model IHA more closely matched theobserved data than Model CHA. The dashed black line in each plot at 0 on the x-axiscorresponds to home teams for whom, on average, the mean of the posterior predictivedistribution matched that shown in our observed data.

For 80% of the teams across all leagues, the posterior predictive distribution usingModel IHA more appropriately reflects the observed data. In MLB, the two modelsperform nearly the same with the exception of the Colorado Rockies, whose home fieldadvantage is underestimated when using Model CHA (see Section 5.3). Discrepenciesin Model IHA offer a slight improvement over those from Model CHA in both the NFLand NHL, with a marked improvement noticed in the NBA. For example, observedhome probabilities for Denver, Utah, and Golden State are underestimated usingModel CHA, while those for Brooklyn, Detroit, New York, and Philadelphia, are, onaverage, overestimated. In the NHL, the posterior predictive distribution using ModelIHA more closely matches the observed data for 25 of the 30 teams.



NFL NHL

MLB NBA

−2 0 2 −1 0 1 2

−1 0 1 −2.5 0.0 2.5 5.00.0

0.1

0.2

0.3

0.4

0.00

0.25

0.50

0.75

1.00

1.25

0.0

0.5

1.0

0.0

0.2

0.4

logit(p)

Den

sity

Fig 2. Posterior predictive distributions. Density curves of 20 posterior predictive distributions oflogit probabilities (in gray) and one curve with the observed distribution of logit probabilities (in red)are overlaid. The bandwith of the density curves is lowered to highlight the jagged nature of sportsbookprices. By and large, the posterior predictive distributions match the observed data.

5. Results. In this section we present our results. We discuss the implicationsof our estimates of team strength and home advantage, as well as the interpretationof our variance and autoregressive parameters. We conclude by evaluating our teamstrength parameters and illustrating how they could be used for predictive purposesand to build league parity metrics.

5.1. Team strength. Table 3 shows summary statistics of the team strength esti-mates, approximated using posterior mean draws for all weeks k and seasons s acrossall four sports leagues. Overall, there tends to be a larger variability in team strengthat any given point in time in both the NFL and NBA, with average posterior coeffi-cient estimates tending to vary between -1.3 and 1.2 in the NBA and -1.0 and 1.0 inthe NFL (on the logit scale) about 95% of the time. For reference, a team-strength of

1.0 on the log-odds scale implies a e1.0

1+e1.0= 73.1% chance of beating a league average

team in a game played at a neutral site. The standard deviation of team strengthis smallest in MLB, suggesting that—relative to the other leagues—team strength ismore tightly packed. Relative to MLB, spread of team strengths are about 1.3, 3.1,



NFL NHL

MLB NBA

−0.050 −0.025 0.000 0.025 −0.050 −0.025 0.000 0.025

Atlanta Hawks

Boston Celtics

Brooklyn Nets

Charlotte Hornets

Chicago Bulls

Cleveland Cavaliers

Dallas Mavericks

Denver Nuggets

Detroit Pistons

Golden State Warriors

Houston Rockets

Indiana Pacers

Los Angeles Clippers

Los Angeles Lakers

Memphis Grizzlies

Miami Heat

Milwaukee Bucks

Minnesota Timberwolves

New Orleans Pelicans

New York Knicks

Oklahoma City Thunder

Orlando Magic

Philadelphia 76ers

Phoenix Suns

Portland Trail Blazers

Sacramento Kings

San Antonio Spurs

Toronto Raptors

Utah Jazz

Washington Wizards

Anaheim Ducks

Arizona Coyotes

Boston Bruins

Buffalo Sabres

Calgary Flames

Carolina Hurricanes

Chicago Blackhawks

Colorado Avalanche

Columbus Blue Jackets

Dallas Stars

Detroit Red Wings

Edmonton Oilers

Florida Panthers

Los Angeles Kings

Minnesota Wild

Montreal Canadiens

Nashville Predators

New Jersey Devils

New York Islanders

New York Rangers

Ottawa Senators

Philadelphia Flyers

Pittsburgh Penguins

San Jose Sharks

St. Louis Blues

Tampa Bay Lightning

Toronto Maple Leafs

Vancouver Canucks

Washington Capitals

Winnipeg Jets

Arizona Diamondbacks

Atlanta Braves

Baltimore Orioles

Boston Red Sox

Chicago Cubs

Chicago White Sox

Cincinnati Reds

Cleveland Indians

Colorado Rockies

Detroit Tigers

Houston Astros

Kansas City Royals

Los Angeles Angels

Los Angeles Dodgers

Miami Marlins

Milwaukee Brewers

Minnesota Twins

New York Mets

New York Yankees

Oakland Athletics

Philadelphia Phillies

Pittsburgh Pirates

San Diego Padres

San Francisco Giants

Seattle Mariners

St Louis Cardinals

Tampa Bay Rays

Texas Rangers

Toronto Blue Jays

Washington Nationals

Arizona CardinalsAtlanta Falcons

Baltimore RavensBuffalo Bills

Carolina PanthersChicago Bears

Cincinnati BengalsCleveland Browns

Dallas CowboysDenver Broncos

Detroit LionsGreen Bay Packers

Houston TexansIndianapolis Colts

Jacksonville JaguarsKansas City ChiefsLos Angeles Rams

Miami DolphinsMinnesota Vikings

New England PatriotsNew Orleans Saints

New York GiantsNew York Jets

Oakland RaidersPhiladelphia EaglesPittsburgh Steelers

San Diego ChargersSan Francisco 49ers

Seattle SeahawksTampa Bay Buccaneers

Tennessee TitansWashington Redskins

Difference (log odds scale)

IHA betterFALSE

TRUE

Fig 3. Posterior predictive distributions by model type. Each dot represents the average differencebetween the posterior predictive distribution and the truth for each team’s home games under theCHA model. The tip of the corresponding arrow represents the same quantity under the IHA model.The difference is smaller under IHA for 80% of the teams.



and 3.6 times wider in the NHL, NFL, and NBA, respectively.

League (q) N* min 2.5th Q1 Q3 97.5th max sd

MLB 9240 -0.553 -0.373 -0.134 0.126 0.337 0.473 0.182NBA 8250 -2.202 -1.268 -0.487 0.477 1.204 1.873 0.660NFL 5440 -1.576 -1.092 -0.402 0.416 1.030 1.906 0.559NHL 9240 -1.034 -0.523 -0.162 0.180 0.438 0.877 0.246

Table 3Summary of average week-level team strength parameters, taken on the log-odds scale. N*: number

of unique team strength draws (teams × seasons × weeks)

Figure 4 shows estimated team strength coefficients over time. Figures 13–16 (shownin the Appendix) provide an individual plot for each sport, which include divisionalfacets to allow easier identification of individual teams. Teams in Figures 4 and 13–16 are depicted using their two primary colors, scraped from http://jim-nielsen.

com/teamcolors/ via the teamcolors package (Baumer and Matthews, 2017) in R.A color key for all teams appears in Figure 17.

As in Table 3, these figures suggest that the NBA and NFL boast larger between-team gaps in quality than the NHL and MLB, implying more competitive balance inthe latter pair of leagues. On one level, this stands somewhat in contrast to competitivebalance as measured using Noll-Scully, which alternatively argues that the NFL ismore competitively balanced than MLB (Berri, 2014). One likely explanation for thisdifference is Null-Scully’s link to number of games played, which artificially makesMLB (162 games) appear less balanced than it actually is and the NFL (16) appearmore balanced. Like Noll-Scully, we conclude that the NBA shows less competitivebalance relative to other leagues.

Our figures also illustrate several other observations. For example, the 2007 NewEngland Patriots of the NFL stand out as having the highest probabilities of beatinga league average team, with an average team strength of 1.91 on the log-odds scale,observed during Week 11. In that season, New England finished the regular season 16-0 before eventually losing in the Super Bowl. The team with the lowest probability ofbeating a league average team is the NBA’s 2007–08 Miami Heat, who during week 23had a posterior mean team strength of -2.2. That Heat team finished with an overallrecord of 15-67, at one point losing 15 consecutive games. Related, it is interesting thatthe team strength estimates of bad teams in the NBA (e.g. the Heat in 2007–08) liefurther from 0 than the estimates for good teams. This possibly reveals the tendencyfor teams in this league to “tank”—a strategy of fielding a weak team intentionallyto improve the chances of having better selection preference in the upcoming playerdraft (Soebbing and Humphreys, 2013).

Another observation is that in the NHL, top teams appear less dominant than adecade ago. For example, there are seven NHL team-seasons in which at least one teamreached an average posterior strength estimate of 0.55 or greater; each of these cameduring or prior to the 2008–09 season. In addition to increased parity, the league’spoint system change in 2005–06—which unintentionally encouraged teams to playmore overtime games (Lopez, 2013)—could be responsible. More overtime contests


http://jim-nielsen.com/teamcolors/

http://jim-nielsen.com/teamcolors/


Fig 4. Mean team strength parameters over time for all four sports leagues. MLB and NFL seasonsfollow each yearly tick mark on the x-axis, while NBA and NHL seasons begin during years labeledby the preceding tick marks.



could lead to different perceptions in how betting markets view team strengths, asovertime sessions and the resulting shootouts are roughly equivalent to coin flips(Lopez and Schuckers, 2016).

As a final point of clarification in Figures 4, 14, and 16, the periods of time withstraight lines of team strength estimates during the 2012–13 season (NHL) and 2011–12 season (NBA) reflect time lost due to lockouts.

5.2. Variance and autoregressive parameters. Table 4 shows the mean and stan-dard deviation of posterior draws for γq,season, γq,week, σq,game, σq,season, and σq,weekfor each q. Before discussing results from these posterior distributions, it is importantto recognize that each variance and autoregressive parameter is uniquely tied to eachsport’s relative logit scale. For example, the average posterior draw of γNBA,season andγMLB,season are both equal to 0.62, implying that relative to each league’s distributionof team strengths, we can expect the same amount of reversion from one season tothe next. However, given that there are larger gaps in the team strengths in the NBA,this corresponds to larger reversions in season-level strength when considered on anabsolute scale.

League (q) γq,season γq,week σq,game σq,season σq,week

MLB 0.618 (0.031) 1.002 (0.002) 0.201 (0.001) 0.093 (0.005) 0.027 (0.001)NBA 0.618 (0.04) 0.977 (0.003) 0.274 (0.002) 0.44 (0.02) 0.166 (0.003)NFL 0.69 (0.042) 0.978 (0.005) 0.233 (0.008) 0.331 (0.019) 0.147 (0.006)NHL 0.542 (0.027) 0.993 (0.003) 0.105 (0.001) 0.121 (0.006) 0.053 (0.001)

Table 4Mean posterior draw (standard deviation) by league.

Posterior draws of σq,game suggest that the highest game-level errors in our log-odds probability estimates occur in the NBA (median posterior draw of σNBA,game= 0.274), followed in order by the NFL, MLB, and the NHL. Interestingly, althoughFigure 4 identifies that the talent gap between teams is smallest in MLB, σMLB,game ≈2×σNHL,game in our posterior draws. We posit that this additional game-level error inMLB is a function of the league’s pitching match-ups, in which teams rotate througha handful of starting pitchers of varying calibers.

We also examine the joint distribution of the variability in team strength on aseason-to-season (σq,season) and week-to-week (σq,week) basis via the contour plot inFigure 18 (Appendix), using separate colors for each q. Figure 18 reveals that thehighest uncertainty with respect to team strength occurs in the NBA, followed inorder by the NFL, NHL, and MLB.

Even when accounting for the larger scale in outcomes, the NBA still stands outas far as increased between-week uncertainty. There are a few possible explanationsfor this. Injuries, the resting of starters, and in-season trades would seemingly havea larger impact in a sport like basketball where fewer players are participating at asingle point in time. In particular, our model cannot precisely gauge team strengthwhen star players who could play are rested in favor of inferior players. Relative tothe other professional leagues, star players take on a more important role in the NBA



(Berri and Schmidt, 2006), an observation undoubtedly known in betting markets.That said, while there is increased variability in our estimate of NBA team strengths,when considering differences in team talent to begin with, these absolute differencesare not as extreme (e.g., a difference in team strength of 0.05 means less in the NBAas far as relative ranking than in the NHL).

Figure 19 (Appendix) displays the joint posterior distribution of γq,season and γq,weekvia contour plots for each q. On a season-to-season basis, team strengths in each of theleagues tend to revert towards the league average of zero as all draws of γq,season < 1for all q. Reversion towards the mean is largest in the NHL (estimated γNHL,season= 0.54, implying 46% reversion), followed by the NBA (38%), MLB (38% reversion),and the NFL (31%). However, the only pair of leagues with non-overlapping credibleintervals are the NFL and NHL. Note that one reason that team strengths may reverttowards zero each year is the structure of each league’s draft, in which newly eligbleplayers are chosen. In expectation, the worst team in each league is most likely to getthe top selection in the following year’s draft, and so by aquiring the best perceivedtalent, those worst teams are more likely to improve. Perhaps one reason that the NFLshows the most consistency over time is that, in general, it is the worst at draftingnewly eligible players (see Lopez (2016) for comparisons in the drafting ability of eachleague).

For each of the NHL, NBA, and NFL, posterior estimates of γq,week (as well as95% credible intervals) imply an autoregressive nature to team strength within eachseason. Interestingly, the NBA and NFL are the least consistent leagues on a week-to-week basis. In MLB, however, team strength estimates quite possibly follow a randomwalk (i.e., γMLB,week = 1), in which the succession of team strength is unpredictable.Alternatively, it is also feasible that MLB team strengths could explode over time(γMLB,week > 1), in which case these estimates would be pulled towards 0 in the longrun (across seasons, via γMLB,season).

Finally, it is worth noting that our estimates for γNFL,week and γNFL,season—0.98and 0.69, respectively—do not substantially diverge from the estimates observed byGlickman and Stern (1998) (0.99 and 0.82). Further, our credible intervals are nar-rower. For example, our 95% credible interval for γNFL,season of (0.61, 0.77) is entirelycontained within the interval of (0.52, 1.28) reported by Glickman and Stern (1998).In fairness, it is unclear if the decreased uncertainty is a function of our model spec-ification (using log-odds of the probability of a win as the outcome, as opposed topoint differential) or because we used a larger sample (10 seasons, compared to 5).

Like Glickman and Stern (1998), we also observe an inverse link in posterior drawsof γNFL,week and γNFL,season. Given that total shrinkage across time is the composite ofwithin- and between-season shrinkage, such an association is not surprising (Glickmanand Stern, 1998). If one source of reversion towards the average were to increase, theother would likely compensate by decreasing.

5.3. The home advantage. Figure 5 shows the 2.5th percentile, median, and 97.5thpercentile draws of each team’s estimated home advantage parameter, presented on



the probability scale. These are calculated by summing draws of αq0 and α(q)i? for alli?. HAs are shown in descending order to provide a sense of the magnitude of differ-ences between the home advantage provided in MLB (league-wide, a 54.0% probabil-ity of beating a team of equal strength at home), NHL (55.5%), NFL (58.9%), andNBA (62.0%). The two franchises that have relocated in the last decade, the AtlantaThrashers (NHL) and Seattle Supersonics (NBA), are also included for the gamesplayed in those respective cities.

Figure 5 depicts substantial between-franchise differences in the home advantagewithin both the NBA and NHL, with lesser between-franchise differences in MLB andthe NFL.

Interestingly, the draws of the home advantage parameters for of a few NFL fran-chises are skewed (see Denver and Seattle, relative to Detroit), potentially the resultof a shorter regular season. Alternatively, the NFL’s HA may vary by season, gametime, or the day of the game. Anecdotally, night games (Thursday, Sunday, or Mon-day) conceivably offer a larger HA than those played during the day (Crabtree, 2014).Informally, NFL team-level HA estimates are similar in effect size to those depictedby Koopmeiners (2012).

In the NBA, Denver (first) and Utah (second) post the best home advantages,with Brooklyn showing the worst. This matches the results of Paine (2013), whofound significantly better performances when comparing Denver and Utah to the restof the league with respect to home and road point differential. In MLB, the ColoradoRockies stand out for having the highest home advantage, while the remaining 29teams boast overlapping credible intervals. We note that teams playing at home inDenver have the largest home advantages in MLB, the NBA, and the NFL, and the7th-highest in the NHL. We speculate that this consistent advantage across sports isrelated to the home team’s acclimation to the city’s notably high altitude.

Differences between teams within the NBA have plausible impacts on league stand-ings. An NBA team with a typical home advantage can expect to win 62.0% of homegames against a like-caliber opponent. Yet for Brooklyn, the corresponding figure is60%, while for Denver, it is 66.1%. Across 41 games (the number each team playsat home), this implies that Denver’s home advantage is worth an extra 1.68 wins ina single season, relative to a league average team. Compared to Brooklyn, Denver’shome advantage is worth an estimated 2.5 wins per year. As one important caveat,our model estimates do not account for varying line-up and injury information. If op-posing teams were to rest their star players at Denver, for example, our model wouldartificially inflate Denver’s home advantage.

As a final note, it is interesting that in comparing leagues, the relative magnitudesof the home advantage match the relative standard deviations in team strength (withthe NBA the highest, followed in order by NFL, NHL, MLB). To check whetheror not the home advantage parameters are independent of team strength estimates(as implied in our model specification), we compared the average posterior draw ofthe home advantage versus the average posterior team strength across all weeks andseasons for each franchise in each sport (plot not shown). Within each sport, there



Philadelphia PhilliesSan Diego PadresCleveland Indians

Los Angeles DodgersCincinnati Reds

Seattle MarinersMiami Marlins

Kansas City RoyalsNew York Mets

Baltimore OriolesLos Angeles Angels

Atlanta BravesWashington Nationals

Chicago CubsDetroit Tigers

San Francisco GiantsPittsburgh PiratesToronto Blue Jays

Chicago White SoxSt Louis Cardinals

Arizona DiamondbacksHouston Astros

Minnesota TwinsTampa Bay Rays

Oakland AthleticsTexas Rangers

New York YankeesBoston Red Sox

Milwaukee BrewersNew York Islanders

Winnipeg JetsOttawa Senators

Colorado RockiesNew Jersey DevilsNew York Rangers

Montreal CanadiensChicago Blackhawks

Philadelphia FlyersToronto Maple LeafsPittsburgh Penguins

Boston BruinsBuffalo Sabres

Arizona CoyotesDallas Stars

St. Louis BluesCarolina Hurricanes

Edmonton OilersTampa Bay Lightning

San Jose SharksFlorida Panthers

Vancouver CanucksWashington Capitals

Detroit Red WingsColorado Avalanche

Los Angeles KingsMinnesota Wild

Columbus Blue JacketsAnaheim Ducks

Nashville PredatorsCalgary Flames

Los Angeles RamsDetroit Lions

Miami DolphinsPhiladelphia EaglesCincinnati BengalsPittsburgh Steelers

New York JetsTennessee TitansIndianapolis ColtsCarolina Panthers

Dallas CowboysOakland Raiders

Green Bay PackersCleveland Browns

Tampa Bay BuccaneersWashington Redskins

Atlanta FalconsSan Diego Chargers

Minnesota VikingsSan Francisco 49ers

Buffalo BillsJacksonville Jaguars

Baltimore RavensNew York Giants

New Orleans SaintsHouston Texans

Chicago BearsKansas City ChiefsArizona CardinalsSeattle Seahawks

New England PatriotsDenver Broncos

Brooklyn NetsDetroit Pistons

New York KnicksPhiladelphia 76ers

Boston CelticsMiami Heat

Toronto RaptorsHouston Rockets

Chicago BullsLos Angeles Lakers

Los Angeles ClippersOrlando Magic

Oklahoma City ThunderMinnesota Timberwolves

Memphis GrizzliesDallas Mavericks

Washington WizardsNew Orleans Pelicans

Cleveland CavaliersMilwaukee Bucks

Indiana PacersAtlanta HawksPhoenix Suns

Charlotte HornetsSan Antonio Spurs

Portland Trail BlazersSacramento Kings

Golden State WarriorsUtah Jazz

Denver Nuggets

0.54 0.620.5890.555

Probability of beating an equal caliber opponent at home

League

MLB

NBA

NFL

NHL

Estimated Home Advantage by Franchise

Fig 5. Median posterior draw (with 2.5th, 97.5th quantiles) of each franchise’s home advantage inter-cept, on the probability scale. We note that the magnitude of home advantages are strongly segregatedby sport, with only one exception (the Colorado Rockies). We also note that no NFL team, nor anyMLB team other than the Rockies, has a home advantage whose 95% credible interval does not containthe league median.imsart-aoas ver. 2014/10/16 file: aoas2017.arxiv.R2.tex date: November 23, 2017


was no obvious link between average team quality and that team’s home intercept, asassessed using scatter plots with a LOESS regression line. That said, further researchmay be needed to precisely define home advantage in light of varying team stregnthestimates, as well game-level characteristics such as time (i.e., afternoon, night) andday (i.e., weekend, weekday.)

5.4. Evaluation of team strength estimates. Ultimately, estimates from Model IHAare designed to estimate team quality at any given point in a season while accountingfor factors such as the home advantage and opponent caliber. If these estimates moreproperly assess team quality than traditional metrics (e.g., won-loss percentage orpoint differential), they should more accurately link to future performance, such ashow well teams will perform over the remainder of the season. Additionally, game-level probabilities estimated from our team strength coefficients should closely trackthe observed money lines.

That said, it is admittedly unfair to use cumulative estimates of team strengthto predict past game outcomes, as future information is implicity used to informthose same game outcomes. In this sense, sequential fits are more appropriate forunderstanding the predictive capability of our state-space models.

Figure 6 shows the coefficient of determination (R2) between each team’s futurewon-loss percentage in a season and each team’s (i) average team strength estimatesfrom sequential Model IHA’s, (ii) season-to-date cumulative point differential, and(iii) season-to-date won-loss percentage. Within each sport, this is computed by gamenumber, which helps to account for league-level differences in season length. For pur-poses of using sequential team strength estimates, we used the mean posterior drawfrom fits that ended the week prior.

Across each sport, our estimates of team strength generally outperform past teamwin percentage and point differential in predicting future win percentage. This gap ismost pronounced earlier in each season, which is not surprising given the instabilityof won-loss percentage and point differential in a small number of games. Differencesremain throughout most of the regular season in MLB, the NHL, and the NFL.However, by the NBA’s mid-season, won-loss ratio and point differential are similarto our estimates of team strength in assessing future performance. By and large, thisconfirms the findings of Wolfson and Koopmeiners (2015), who identified that mostof the information needed to predict the remainder of the NBA season is containedwithin the first third of the year.

As a second check of predictive accuracy, we compare these predicted game-levelprobabilities to known game outcomes. Table 5 highlights the area under the receiveroperating characteristic curve (AUC), which calculates the expectation that a ran-domly drawn probability from a winning home team is greater than a randomly drawnprobability of a losing home team (higher is better). Also included is the Brier score(lower is better), along with an accompanying p-value as implemented for calibrationaccuracy in Spiegelhalter (1986).

For each of the NBA, NFL, and NHL, AUC and Brier metrics suggest that predic-



NFL NHL

MLB NBA

4 8 12 16 0 20 40 60 80

0 50 100 150 0 20 40 60 80

0%

20%

40%

60%

80%

0%

20%

40%

60%

80%

Game of season

Type

Point differential

Sequential

Win %

Coefficient of determination with future in−season win %

Fig 6. Coefficient of determination with future in-season win percentage. We note the improvementour team strength estimates offer over season-to-date win percentage and season-to-date point dif-ferential in most sports, especially early in the season. R2 values tend to 0 as the number of futuregames goes to 0.



AUC Brier ScoreLeague (q) observed sequential observed (p-value) sequential (p-value)

MLB 0.605 0.573 0.241 (0.996) 0.245 (0.333)NBA 0.756 0.756 0.194 (0.803) 0.194 (0.759)NFL 0.682 0.685 0.226 (0.34) 0.226 (0.548)NHL 0.595 0.589 0.242 (0.88) 0.243 (0.486)

Table 5AUC values and Brier scores (p-values) by sport. Observed probabilities use known probabilities frombetting markets, while sequential probabiltiies use predictions from posterior draws using sequential

fits of Model IHA.

tions made from sequential fits can closely approximate the observed game probabil-ities. However, our predictions yield a lower AUC and a higher Brier score in MLB,which likely reflects our inability to account for each game’s starting pitcher.

Although results from these predictions do not suggest an existance of an arbi-trage opportunity (recall that sports books add a vig to each team’s price), they doimply that both our team strength and home advantage estimates can be used to ex-tract accurate game-level projections. Further, that there is no major deviation fromthe observed data is comforting with respect to our choice of a model for the gameprobabilities.

5.5. How often does the best team win? A new measure of league parity. We con-clude by addressing our initial question about the inherent randomness of game out-comes.4

One simple way to compare league randomness would be to contrast the observeddistribution of p(q,s,k)ij ’s between each q. However, while sportsbook odds can beused to infer the probability of each team winning, these odds are only providedfor scheduled games. As a result, any between-league comparisons using sportsbookodds alone would be contingent upon each league’s actual schedule, and they maynot accurately reflect differences that would be observed if all teams were to play oneanother.

A second option would be to contrast our posterior draws of θ(q,s,k)i for all i, eitheracross time periods or at a fixed point in time, as these estimates account for leagueparticulars such as strength of schedule. However, such a procedure would not scaleto other sports or leagues where betting market data may be unavailable. Rather,we would prefer a metric that can be applied generally to any competitive scenariowhere paired comparison probabilities can be calculated.

To assess the equivalence of all teams in each league, we consider the likelihoodthat—given any pair of teams chosen at random—the better team wins, by simu-lating estimates of p(q,s,k)ij using posterior draws of team strength, home advantage,and game level error. For our purposes, we define the better team to be the one, apriori, with a higher probability of winning that game. If a contest has no inherent

4Our approach here is not unlike that of James, Albert and Stern (1993).



randomness (consider the Harlem Globetrotters), then the better team always wins.5

Conversely, if game-level variability is large relative to the difference in team strength,then even the inferior team might win nearly half the time.

Using our posterior draws, we approximate the distribution of game-level probabil-ities between two randomly chosen teams using the following steps. Posterior drawsfrom Model IHA are used.

Given sport q with season length Kq, number of seasons Sq, and number of teamstq,

1. Draw season s from {1, . . . , Sq}, and week k from {1, . . . ,Kq}.2. Draw teams i and j from {1, . . . , tq} without replacement.3. Sample one posterior draw of team strength for i and j, θ(q,s,k)i and θ(q,s,k)j ,

respectively, from the posterior distributions of i and j’s team strength estimatesduring season s at week k. For simplicity, assume θ(q,s,k)i > θ(q,s,k)j .

4. Sample one posterior draw of the HA, αq0 , from the posterior distribution ofαq0 , as well as one posterior draw of team i’s home advantage, α(q)i∗.

5. Sample one posterior draw of the game-level variance parameter, σ2q,game, anddraw a game-level error, εq,game, from εq,game ∼ N(0, σq,game)

6. Impute the simulated log-odds of i beating j, logit(p(q,s,k)ij) = αq0 + α(q)i∗ +

θ(q,s,k)i − θ(q,s,k)j + εq,game.

7. Transform logit(p(q,s,k)ij) into a probability to obtain a simulated estimate,pq,sim, where pq,sim = p(q,s,k)ij

8. Repeat the above steps nsim times to obtain pq = {pq,1, . . . , pq,nsim}.

For each q, we simulated with nsim = 1000. Additionally, to remove the effect ofeach league’s HA on simulated probabilities, we repeated the process fixing αq0 =α(q)i∗ = 0 for each league to reflect game probabilities played in absence of a homeadvantage.

Figure 7 shows the cumulative distribution functions (CDFs) for each set of proba-bilities in each league. The median probability of the best team winning a neutral sitegame is highest in the NBA (67%), followed in order by the NFL (64%), NHL (57%),and MLB (56%). The spread of these probabilities are of great interest. Nearly everysimulated MLB and NHL game played at a neutral site is less than a 3:1 propositionwith respect to the best team winning (75%). Meanwhile, roughly 27% of NBA and20% of NFL neutral site match-ups are greater than this 3:1 threshold.

Factoring in each league’s home advantage works to exaggerate league-level differ-ences. When the best team plays at home in the NBA, it is always favored to win atleast 60% of the time, with the middle 50% of games ranging from a 68% probabilityto an 84% probability. Meanwhile, even with a home advantage, it is rare that thebest MLB team is ever given a 70% probability of winning, with the middle 50% ofgames ranging from 57% to 63%.

5The Harlem Globetrotters are an exhibition basketball team that plays hundreds of games in ayear, rarely losing.



All games

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Fig 7. Cumulative distribution function (CDF) of 1000 simulated game-level probabilities in eachleague, for both neutral site and home games, with the better team (on average) used as the referenceand given the home advantage.



Finally, we use the CDFs displayed in Figure 7 to quantify the cumulative differencebetween each league’s game-level probabilities and a league of coin flips by estimatingthe approximate area under each curve. Let RegParityq be our regular season paritymeasure, such that

RegParityq = 2

∫ 1

0.5P (pq ≤ x)dx ,

where we multiply by 2 in order to scale so that 0 ≤ RegParityq ≤ 1, where 1represents complete parity (every game a coin flip) and 0 represents no parity (everygame outcome pre-determined).

For games with no home advantage, RegParityMLB = 0.87, followed by the NHL(0.84), NFL (0.70), and NBA (0.66). When the best team has a home advantage,parity is again the greatest in the MLB (0.79), followed by the NHL (0.73), NFL(0.55), and NBA (0.47). These results suggest that when the best team is playing athome, the NBA is closer to a world where every game outcome is predetermined thanto one where every game outcome is a coin flip. Meanwhile, even when giving the bestteam a HA, MLB game outcomes remain lightly-weighted coin flips.

5.5.1. Parity in postseason tournaments. Notions of parity in the regular seasoninfluence which teams make the playoffs, but each league conducts a single-eliminationpostseason tournament with a different structure. To what extent do those structuresmitigate or reinforce the parity levels discussed in the previous section? We addressthese questions using our team strength estimates.

First, we collect the z ∈ {8, 16} teams with the highest average team strengthestimates over the last four weeks of each season, in each sport. We then seed (indescending order of team strength, irrespective of division or conference) and simulate1000 postseason tournaments, in which each round consists of a best-of-7-game series,with the higher seed having the home field advantage in each round. The resultsshown in Figure 20 (in the Appendix) confirm that the relationship between seed andtournament finish is strongest in the NBA and the NFL, and considerably weaker inMLB and the NHL. These findings accord with our regular season parity measures.

Next, we construct a postseason tournament parity metric that acts as a pseudo-R2. Let F = (F1, F2, . . . , Fz) be a z-dimensional random vector with the dth elementindicating the round of tournament finish of the dth seed. 6 That is, for the dth-seededteam, Fd = 1 indicates that team finished as tournament champions, Fd = 2 impliesthat team finished as runners-up, and so on. In a z-team tournament in which thehigher seeded team always wins (i.e. the seeds determine the finish), the vector F isconstant with F1 = 1, F2 = 2, F3 = F4 = 3, F5 = F6 = F7 = F8 = 4, etc., and ingeneral, E[Fd] = Fd = dlog2 d + 1e for d = 1, 2, . . . , z. In the other extreme, wherethe seed is irrelevant (i.e., all values of θ are equal and there is no home advantage),E[Fd] =

∑zd=1

1z · dlog2 d+ 1e = fz, where fz is a constant that depends on z.

6We note that F depends on the vector of team strengths.



We define a pseudo-R2 as:

PostParityz = 1− (E[F]− fz1z)′(E[F]− fz1z)∑zd=1(dlog2 d+ 1e − fz)2

,

where d = 1, . . . , z iterates over the seeds, fz is the seed-weighted expected finishround (e.g., 4.0625 for a 16-team tournament), and 1z is a vector of ones of lengthz. A PostParityz value of 0 indicates that the higher seed always wins, while aPostParityz value of 1 occurs when all seeds have the same expected finish. In a16-team, 7-game series tournament, the NBA and NFL’s PostParity16 values (0.43and 0.51) lag far behind those of MLB and the NHL (0.88 and 0.85, respectively).

While these simulations force all sports to use the same postseason tournamentformat, reality is quite different. Accordingly, we simulate tournaments while varyingthe number of teams who qualify (8 or 16) as well as the length of each postseasonseries (selected odd numbers between 1 and 75). Figure 8 allows us to compare val-ues of PostParityz for different tournament structures across all four sports. WhilePostParity8 and PostParity16 values may not be directly comparable, we note thatthe 1-game series played in the NFL results in parity similar to the current MLB andNHL formats. This leaves the NBA alone as the sport whose postseason tournamentmost likely coronates the strongest regular season teams. Conversely, the playoff struc-ture in MLB, which includes a single-game wild card play-in7 and a 5-game divisionseries, serves to undermine advantages conferred based on seed. In order to approachthe level of parity (or lack thereof) of the NBA playoffs, MLB would have to switchto a 16-team tournament in which each round was approximately 75-game series.Conversely, in order to the achieve the level of parity in the other three sports, theNBA would have to reduce the number of playoff teams to 8, and play a single-gametournament.

Postseason parity cuts both ways: a tournament in which the higher seeds alwayswin is potentially less interesting, but a tournament in which seeds don’t mattermight compromise the competitiveness of late-season games for playoff teams. Thisrepresents a philosophical choice for commissioners. The NBA has clearly chosen apostseason structure that—relative to other sports—largely ensures that the bestteams will win most of the time. We suspect that this arrangement is comfortingfor players and team executives, since the hard work of building a good team isremunerated with postseason success. On the other hand, early-round games maysuffer from lack of interest, since fans may consider the outcomes predetermined.Conversely, MLB (and to a slightly lesser extent the NFL and NHL) postseasonstructure serves to maximize fan interest (recall the outcome uncertainty hypothesis),while offering few postseason rewards (other than entry) for regular season success.This may be an acceptable trade-off, since the regular season is so long and relativelyfew teams make the playoffs. Still, it may be profoundly frustrating to players andteam executives.

7We did not include the wild card game in our simulations.



8 16

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Fig 8. Parity measures for simulated playoff tournaments. Each line shows how our pseudo-R2 paritymetric changes as a function of tournament series length for both 8- and 16-team tournaments ineach sport. We note that in order for MLB to achieve the same lack of parity as the NBA, it wouldhave to play 75-game series in a 16-team tournament. Conversely, the NBA would have to switch toan 8-team, single-game tournament to match the parity of the other three sports.

6. Conclusion.

6.1. Summary. We propose a modified Bayesian state-space framework that canbe used to estimate both time-varying strength and variance parameters in order tobetter understand the underlying randomness in competitive organizations. We applythis model to the NBA, NFL, NHL, and MLB.

Our first finding relates to the relative equivalence of the four leagues. At a singlepoint in time, team strength estimates diverge substantially more in the NBA andNFL than in the NHL and MLB. In the latter two leagues, contests between tworandomly chosen teams are closer to a coin-flip, in which each team has a reasonableshot at winning. Understanding this underlying randomness would appear to be cru-cial for decision makers in these leagues. At critical moments in a team’s evolution,such as the a trade deadline, free agency period, or the decision to fire a coach, werecommend that team officials look past wins and losses to better understand teamstrength in the context of their league. As one easy example, it is insufficient to eval-uate a baseball or hockey team based on their performance in the postseason alone,given that so many of those contests are nearly 50-50 outcomes.

Our next finding relates to the relative equivalence of the home advantage in eachleague, with the NBA well ahead of the pack, with teams averaging a 62.0% chance ofwinning versus a like-caliber opponent. We also show that the home advantage variesmost significantly between venues within each of the NBA and the NHL. In the NBA,for example, the league’s best team home advantage is worth a few wins per year, in



expectation, over the league’s worst home advantage. Moreover, with the exceptionof the Colorado Rockies, it is not clear that any MLB or NFL team has a statisticallysignificant home effect.

Finally, we identify that team strengths derived from sequential fits are nearlyas accurate for predictive purposes as the observed game probabilities, as judgedby both links to future team performance and game-level outcomes. We concludeby using team strength draws to propose two parity metrics, one for regular seasoncomparisons and another for postseason contrasts.

6.2. Discussion. There are several options for applying or extending our model.Generally, the conditions needed to apply our framework are minimal; only pairedevents, outcome probabilities, and some unit of time are needed.

As alternative examples in sports, comparisons between divisions of teams in thesame organization (as in English soccer) or between the top leagues of the samesport (as in European soccer) would follow a similar structure to the one provided.Alternatively, in any sporting league, modeling the impact of structural changes (suchas free agency, expansion or scoring system updates) would be straightforward to testby adding covariates to our models. Note that team sports are not required for ourmodel to apply: a similar framework could assess the caliber of tennis players, whoserelative strengths fluctuate over time both within and across seasons. Competitivebalance questions within amateur sport (for example, conferences in NCAA football,or even across all intercollegiate sports) would follow a similar design.

There are also several ways our model could generalize to other competitive spheresof life. Assessing player and team strength in the increasing popular (and visible)world of online gaming could be a future application. Online trivia leagues (e.g. theLearned League) also pit players organized into divisions by ability in head-to-headcompetition—their relative strengths could be modeled in our framework. Given thatpolitical elections have only one outcome, traditional prediction models are difficultto judge and calibrate. However, since our framework does not require outcomes, andexpansive betting market data that tracks candidates’ probabilities over time exists,applying our models to political elections is another possible extension. Comparisonsin the volitility of candidate support over time, either between states, countries, orelection cycles, may be feasible.

Additionally, researchers of the NBA, NFL, NHL, and MLB could explore severalhypotheses using our provided team strength estimates. One option would be to testhow each league’s scheduling quirks impact won-loss standings. For example, what isthe consequence of the unbalanced schedule used in the NFL, relative to a balanceddesign? Given each league’s schedule, how likely is it for the best team to qualifyfor postseason play? Finally, one could use time-varying estimates of team strengthto consider the existence of tanking, in which teams—in order to secure a betterdraft position—are better off losing games later in the season. While this has beendemonstrated in basketball using betting market data (Soebbing and Humphreys,2013), it would also be worth looking at tanking in other leagues, or if team interest



in tanking corresponds to the perceived talent available in the upcoming draft.Opportunities to improve our model are also plentiful. Both Models IHA and CHA

make use of the logit transform with game-level probabilities. Although our posteriorpredictive checks do not seem to indicate that this assumption is unjustified, alterna-tive transformations may be considered. We implemented a version of Model IHA thatused the arcsin-sqrt transformation—the variance-stabilizing transformation for bino-mial proportions—and found nearly identical results with both game-level probabilitypredictions and team strength estimates.

To estimate predictions from the sequential fits, we repeatedly applied our MCMCalgorithm in each week. In place, sequential Monte Carlo samplers (Gilks and Berzuini,2001; Del Moral, Doucet and Jasra, 2006) would have been more efficient. In the sportsof soccer and hockey, one improvement would consider three-way lines that includethe probability of a tie game. Specifically, soccer betting markets use a vector ofprobabilities (win, loss, tie). To account for these complexities, Firth (2017) proposeda generalized Bradley-Terry model to simultaneously model both wins and draws, onethat could likewise start with imputed game probability vectors. Finally, a comparisonof team strengths estimated by our model, as well as those fit by Glickman and Stern(1998) and Koopmeiners (2012), could more acutely identify the impact of usingbetting market data relative to point differential and won-loss outcomes.

To maintain consistency with the NFL’s calendar, we considered time on a weeklybasis; more refined approaches may be appropriate in other sports. As an example,investigation into starting pitchers in baseball—who change daily—could lead to novelfindings. Additionally, another model specification could consider the possibility thattime-varying estimates of team strength follow something other than an autoregressivestructure. One alternative assumption, for example, is a stochastic volatility process(Glickman, 2001). In this respect, our model can be considered a starting point forthose looking to dig deeper in any sport without losing an ability to make cross-leagueor cross-sport comparisons.

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Supplementary Materials for

“A unified approach to understanding randomness insport”



E-mail: [email protected] E-mail: [email protected]

E-mail: [email protected]


mailto:[email protected]




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Fig 12. Trace plots of NHL parametersimsart-aoas ver. 2014/10/16 file: aoas2017.arxiv.R2.tex date: November 23, 2017


Fig 13. Team strength coefficients over time for Major League Baseball.



Fig 14. Team strength coefficients over time for the National Basketball Association.



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2010

2011

2012

2013

2014

2015

2016

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

−1

0

1

2

−1

0

1

2

−1

0

1

2

−1

0

1

2

Season

Team

Str

engt

h P

aram

eter

(lo

g−od

ds s

cale

)Team strength parameters over time, NFL

Fig 15. Team strength coefficients over time for the National Football League.



Fig 16. Team strength coefficients over time for the National Hockey League.



Fig 17. Team-color mappings used throughout the paper. (Baumer and Matthews, 2017)



Fig 18. Contour plot of the estimated season-to-season and week-to-week variability across all fourmajor sports leagues. By both measures, uncertainty is lowest in MLB and highest in the NBA.

Fig 19. Contour plot of the estimated season-to-season and week-to-week autoregressive parametersacross all four major sports leagues.



perf

ect p

arity

pred

estin

atio

nP

seud

o−R

^2=

0.8

81

perf

ect p

arity

pred

estin

atio

nP

seud

o−R

^2

= 0

.516

perf

ect p

arity

pred

estin

atio

nP

seud

o−R

^2=

0.4

34

perf

ect p

arity

pred

estin

atio

nP

seud

o−R

^2

= 0

.844

NF

LN

HL

MLB

NB

A

12

48

161

24

816

12345 12345

Tour

nam

ent S

eed

(d)

Tournament Finish (Fd, round)

2500

5000

7500

1000

0

N

Sim

ulat

ed 1

6−Te

am T

ourn

amen

ts, 2

006−

2016

Fig 20. Relationship between seed and finish in simulated 16-team, 7-game series playoff tournaments.One thousand tournaments were simulated for each sport in each year. The horizontal dotted grayline represent how the tournaments would play out with perfect parity, while the stepped gray linerepresents tournaments that play out in perfect accordance with seed.


Documents

Submitted to the Annals of Applied Statistics